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JEE Main202125 Feb 2021Evening ShiftMathematicsParabolaActual

A line is a common tangent to the circle x - 3 2 + y 2 = 9 and the parabola y 2 = 4 x . If the two points of contact a , b and c , d are distinct and lie in the first quadrant, then 2 a + c is equal to ___ .

Correct answer

0

Step-by-step solution

Let coordinate of point A t 2 , 2 t    ∵ a = 1 equation of tangent at point A y t = x + t 2 x - t y + t 2 = 0 centre of circle 3 , 0 Now Perpendicular distance = radius 3 - 0 + t 2 1 + t 2 = 3 3 + t 2 2 = 9 1 + t 2 9 + t 4 + 6 t 2 = 9 + 9 t 2 t = 0 , - 3 , 3 So point A 3 , 2 3 ⇒ a = 3 ,   b = 2 3 (Since it lies in first quadrant) For point B which is foot of perpendicular from centre 3 , 0 to the tangent x - 3 y + 3 = 0 c - 3 1 = d - 0 - 3 = - 3 - 0 + 3 4 ⇒ c = 3 2    d =

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